Calculators / Calculus 3 / Angle Between a Line and a Plane (3D) Calculator

Angle Between a Line and a Plane (3D) Calculator

Published on: June 7, 2026

This Angle Between a Line and a Plane (3D) Calculator helps you find the angle between a line and a plane in three-dimensional space. Use the line’s direction vector and the plane’s normal vector, then apply the angle formula to calculate the result. This shows how the line is inclined relative to the plane. It is a simple way to check answers, understand the method clearly, and practise 3D coordinate geometry step by step.

Step-by-step method

  1. Identify the line direction vector v and the plane normal vector n.
  2. Compute the dot product |v · n|.
  3. Compute the magnitudes |v| and |n|.
  4. Use sin(α) = |v · n| / (|v||n|) and solve for α.

Formula:

Ax + By + Cz = d
n = <A, B, C>
sin(α) =
|v · n|
|v||n|

Example 1: <2,-1,4>; 2x-3y+4z=10

Step 1 - Identify v (line direction) and n (plane normal).

In this problem: For Ax+By+Cz=d, the normal is n=<A,B,C>. The line gives v=<a,b,c>.

v = < 2, -1, 4 >
n = < 2, -3, 4 >

Step 2 - Compute the dot product magnitude |v·n|.

In this problem: Compute v·n and then take absolute value.

|v · n| = |2·2 + -1·-3 + 4·4| = |23| = 23

Step 3 - Compute magnitudes |v| and |n|.

In this problem: Use |v| = √(a²+b²+c²) and |n| = √(A²+B²+C²).

|v| = √22 + -12 + 42 = √21
|n| = √22 + -32 + 42 = √29

Step 4a - Substitute solved values into sin(α) = |v·n| / (|v||n|).

In this problem: Use the already-computed |v·n|, |v|, and |n| values.

sin(α) =
|v · n|
|v||n|
=
23
√21 √29

Step 4b - Solve for α.

In this problem: Take arcsin to get the angle between the line and the plane.

α = arcsin(
23
√21 √29
) ≈ 68.75° (≈ 1.20 rad)

Final answer: Angle = 180*asin(23*sqrt(609)/609)/pi degrees

Example 2: <1,2,-2>; 2x-y+2z=7

Step 1 - Identify v (line direction) and n (plane normal).

In this problem: For Ax+By+Cz=d, the normal is n=<A,B,C>. The line gives v=<a,b,c>.

v = < 1, 2, -2 >
n = < 2, -1, 2 >

Step 2 - Compute the dot product magnitude |v·n|.

In this problem: Compute v·n and then take absolute value.

|v · n| = |1·2 + 2·-1 + -2·2| = |-4| = 4

Step 3 - Compute magnitudes |v| and |n|.

In this problem: Use |v| = √(a²+b²+c²) and |n| = √(A²+B²+C²).

|v| = √12 + 22 + -22 = 3
|n| = √22 + -12 + 22 = 3

Step 4a - Substitute solved values into sin(α) = |v·n| / (|v||n|).

In this problem: Use the already-computed |v·n|, |v|, and |n| values.

sin(α) =
|v · n|
|v||n|
=
4
√9 √9

Step 4b - Solve for α.

In this problem: Take arcsin to get the angle between the line and the plane.

α = arcsin(
4
√9 √9
) ≈ 26.39° (≈ 0.46 rad)

Final answer: Angle = 180*asin(4/9)/pi degrees